Square Root of 642

√642 at a glance

Exact value

√642

Decimal (10 places)

25.3377189186

Rounded

25.3 · 25.34 · 25.338

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.337719

Prime factorization

2 × 3 × 107

Cube root

8.626706

How to simplify √642

The prime factorization of 642 is 2 × 3 × 107. Every prime appears only once, so there is no pair to bring outside the radical — √642 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 642, 2, 3 and 107 appear an odd number of times, so √642 is irrational and 25.3377189186 is a rounded value.

Where √642 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √642 lies between 25 and 26. 642 is 17 above 625 and 34 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.3333 (0.02% low)
• Tangent from 25, i.e. 25 + 17 ÷ 50: 25.3400 (0.01% high)
• Tangent from 26, i.e. 26 − 34 ÷ 52: 25.3462 (0.03% high)

For √642 the tangent at 25 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 642 is just 17 above 625.

Finding √642 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √642 = 25.3377189186 to every decimal shown.

√642 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √642 the pattern is [25; 2, 1, 24, 1, 2, 50] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √642 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 642y² = 1. Its smallest solution in positive whole numbers is x = 5,777, y = 228.

√642 in geometry and everyday measurements

• A square garage floor of 642 square feet measures about 25.34 ft (25 ft 4 in) per side, and its corner-to-corner diagonal is √1284 ≈ 35.8 ft.
• 642 is not a sum of two whole-number squares — the prime factor 3 and 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √642 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 25 box, because 1² + 4² + 25² = 642.

Square roots near √642 and related roots

• The cube root of 642 is about 8.626706.
• Squaring undoes the root: (√642)² = 642, while 642² = 412,164 — the number whose square root is 642.

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